Trajectory-Based Nonadiabatic Molecular Dynamics Reminder From

Outline

1 Ab initio molecular dynamics Why Quantum Dynamics? Nuclear quantum e↵ects

2 Mixed quantum-classical dynamics Trajectory-based nonadiabatic molecular dynamics Ehrenfest dynamics A TDDFT approach Adiabatic Born-Oppenheimer dynamics Nonadiabatic Bohmian dynamics Trajectory Surface Hopping Ivano Tavernelli 3 TDDFT-based trajectory surface hopping IBM -Research Zurich Nonadiabatic couplings in TDDFT

4 TDDFT-TSH: Applications Summer School of the Photodissociation of Oxirane Max-Planck-EPFL Center for Molecular Nanoscience & Technology Oxirane - Crossing between S1 and S0

5 TSH with external time-dependent ﬁelds Local control (LC) theory Application of LCT

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Ab initio molecular dynamics

1 Ab initio molecular dynamics Reminder from last lecture: potential energy surfaces Why Quantum Dynamics? Nuclear quantum e↵ects

2 Mixed quantum-classical dynamics Ehrenfest dynamics Adiabatic Born-Oppenheimer dynamics Nonadiabatic Bohmian dynamics Trajectory Surface Hopping

3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT

4 TDDFT-TSH: Applications Photodissociation of Oxirane

Oxirane - Crossing between S1 and S0

5 TSH with external time-dependent ﬁelds Local control (LC) theory Application of LCT We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei...

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Ab initio molecular dynamics Ab initio molecular dynamics Why Quantum Dynamics? Reminder from last lecture: potential energy surfaces Why Quantum dynamics?

GS adiabatic dynamics (BO vs. CP)

BO MI R¨ I (t)= min⇢ EKS ( i [⇢] ) r { }

Energy ¨ CP µi i (t) = EKS ( i (r) )+ constr. | i h i | { } h i | { } MI R¨ I (t)= EKS ( i (t) ) r { }

ES nonadiabatic quantum dynamics

Wavepacket dynamics (MCTDH) Trajectory-based approaches - Tully’s trajectory surface hopping (TSH) - Bohmian dynamics (quantum hydrodyn.) - Semiclassical (WKB, DR) - Path integrals (Pechukas)

Energy - Mean-ﬁeld solution (Ehrenfest dynamics) Density matrix, Liouvillian approaches, ... We have electronic structure methods for electronic ground and excited states... Now, we need to propagate the nuclei...

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Why Quantum Dynamics? Ab initio molecular dynamics Why Quantum Dynamics? Why Quantum dynamics? Why Quantum dynamics?

GS adiabatic dynamics GS adiabatic dynamics First principles Heaven First principles Heaven

Energy Ab initio MD with WF methods Energy Ab initio MD with DFT & TDDFT [CP] Ab initio MD with WF methods classical MD Ab initio MD with DFT & TDDFT [CP] Coarse-grained MD classical MD ... Coarse-grained MD ... No principles World No principles World ES nonadiabatic quantum dynamics ES nonadiabatic quantum dynamics First principles Heaven (-) We cannot get read of electrons Ab initio MD with WF methods Ab initio MD with DFT & TDDFT [CP] (-) Nuclei keep some QM ﬂavor (-) Accuracy is an issue # Energy Models Energy (-) Size can be large (di↵use excitons) ? # (+) Time scales are usually short (< ps) No principles World

1 Ab initio molecular dynamics Nonadiabatic e↵ects requires quantum nuclear dynamics Why Quantum Dynamics? Nuclear quantum e↵ects The nuclear dynamics cannot be described by a single classical trajectory (like in the ground state -adiabatically separated- case) 2 Mixed quantum-classical dynamics Ehrenfest dynamics Adiabatic Born-Oppenheimer dynamics Nonadiabatic Bohmian dynamics Trajectory Surface Hopping

3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT

4 TDDFT-TSH: Applications Photodissociation of Oxirane

Oxirane - Crossing between S1 and S0

5 TSH with external time-dependent ﬁelds Local control (LC) theory Application of LCT

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Multi-trajectories vs. wavepacket Wavepacket-dynamics vs. trajectory-based QD

Trajectory-based dynamics Wavepacket-based dynamics Trajectory-based dyn. Wavepacket dyn.

dynamics in phase space reaction coordinates d=3N-6 (unbiased) d=1-10 (knowledge based)

on-the-ﬂy dynamics pre-computed energy surfaces on a grid good scaling (linear in d) scaling catastrophe (Md ) coupling to real environment coupling to harmonic bath slow convergence good convergence Dynamics in 3N dimensions Dynamics in 8dimensions approximate ‘exact’ in low dimensions Approximated quantum dynamics ’Exact’ quantum dynamics (nuclear (trajectory ensemble) wavepakets)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Wavepacket dynamics with MCTDH Wavepacket dynamics with MCTDH

The multiconﬁguration time-dependent Hartree , MCTDH, approach. Potential energy surfaces computed along 2 Precomputed TDDFT surfaces along vibrational modes (Spin Orbit Coupling (SOC) selected modes (8). 1) collecting variables: described by the Breit-Pauli operator.)

(R , R ,...,R ) (Q ,...,Q ) 1 2 3N ! 1 M state sym. eV f S1 B3 2.43 0.0006 with M << 3N,and S2 A 2.51 0.0000 S3 B3 2.69 0.1608 S4 B1 2.83 0.0023 Q1(R1, R2,...,R3N ) state sym. eV f T1 A 2.27 0.0000 T2 A 2.31 0.0000 ... T3 B3 2.33 0.0000 T4 B3 2.36 0.0000 QM (R1, R2,...,R3N ) 1 mode cm sym. type 19 193.61 B3 Rocking 2) Ansatz: linear combination of Hartree products 21 247.61 B3 Jahn Teller 31 420.77 B3 As. Stretching 41 502.65 B3 Rocking n1 nf f 55 704.75 B3 Rocking (k) 58 790.76 B3 Rocking SOC at equilibrium: (Q1, Q2,...,QM )= Aj ...j (t) (Qk , t) 25 270.85 A Twisting 1 1 1 f jk S1/T1 =65cm ; S2/T1 =35cm ··· SOC at distorted geometry: jX1=1 jXf =1 kY=1 1 1 S1/T1 =30cm ; S2/T1 =30cm 3) EOF from Dirac-Frenkel variational principle: Hˆ i @ =0. h | @t | i Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Wavepacket-dynamics of Cu-phenanthroline Wavepacket-dynamics of Cu-phenanthroline

0fs 20 fs

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Wavepacket-dynamics of Cu-phenanthroline Wavepacket-dynamics of Cu-phenanthroline

100 fs 500 fs

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Trajectory-based quantum dynamics Trajectory-based quantum dynamics: an example

Ultrafast tautomerization of 4-hydroxyquinoline (NH ) · 3 3 Hydrogen or proton transfer?

CIS/CASSCF for the in-plane geometry ⇡⇡ /⇡ crossing leads to a hydrogen atom transfer. ! ⇤ ⇤ S. Leutwyler et al.,Science,302,1736(2003)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Trajectory-based quantum dynamics: an example Trajectory-based quantum dynamics: an example

Ultrafast tautomerization of 4-hydroxyquinoline (NH ) Ultrafast tautomerization of 4-hydroxyquinoline (NH ) · 3 3 · 3 3 Hydrogen or proton transfer? Hydrogen or proton transfer?

What about TDDFT combined with nonadiabatic dynamics (1)? What about TDDFT combined with nonadiabatic dynamics (2)?

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Ab initio molecular dynamics Nuclear quantum e↵ects Ab initio molecular dynamics Nuclear quantum e↵ects Trajectory-based quantum dynamics: an example Trajectory-based quantum dynamics: an example

Ultrafast tautomerization of 4-hydroxyquinoline (NH ) Ultrafast tautomerization of 4-hydroxyquinoline (NH ) · 3 3 · 3 3 Hydrogen or proton transfer? Hydrogen or proton transfer? Similar observations with CASPT2 calculations: With TDDFT, we observe:

Symmetry breaking.

No crossing with the ⇡⇤ state. Proton transfer instead of a hydrogen transfer.

Guglielmi et al.,PCCP,11,4549(2009).

Forcing in-plane symmetry: hydrogen transfer. Unconstrained geometry optimization leads to a proton transfer! Fernandez-Ramos et al.,JPCA,111,5907(2007).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Ab initio molecular dynamics Nuclear quantum e↵ects Mixed quantum-classical dynamics

Summary: Why trajectory-based approaches? 1 Ab initio molecular dynamics Why Quantum Dynamics? W1 In “conventional” nuclear wavepacket propagation potential energy surfaces are needed. Nuclear quantum e↵ects W2 Diculty to obtain and fit potential energy surfaces for large molecules. 2 Mixed quantum-classical dynamics W3 Nuclear wavepacket dynamics is very expensive for large systems (6 degrees of freedom, 30 Ehrenfest dynamics for MCTDH). Bad scaling. Adiabatic Born-Oppenheimer dynamics T1 Trajectory based approaches can be run on-the-fly (no need to parametrize potential Nonadiabatic Bohmian dynamics energy surfaces). Trajectory Surface Hopping T2 Can handle large molecules in the full (unconstraint) configuration space. 3 T3 They o↵er a good compromise between accuracy and computational e↵ort. TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT

4 TDDFT-TSH: Applications Photodissociation of Oxirane

Oxirane - Crossing between S1 and S0

5 TSH with external time-dependent ﬁelds Local control (LC) theory Application of LCT

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Mixed quantum-classical dynamics Starting point Tarjectory-based quantum and mixed QM-CL solutions

We can “derive” the following trajectory-based solutions: The starting point is the molecular time-dependent Schr¨odingerequation:

@ Nonadiabatic Ehrenfest dynamics dynamics i~ (r, R, t)=Hˆ (r, R, t) @t I. Tavernelli et al., Mol. Phys., 103,963981(2005).

where Hˆ is the molecular time-independent Hamiltonian and (r, R, t)thetotalwavefunction (nuclear + electronic) of our system. Adiabatic Born-Oppenheimer MD equations

In mixed quantum-classical dynamics the nuclear dynamics is described by a swarm of classical Nonadiabatic Bohmian Dynamics (NABDY) trajectories (taking a ”partial” limit ~ 0forthenuclearwf). ! B. Curchod, IT, U. Rothlisberger, PCCP, 13,32313236(2011) In this lecture we will discuss two main approximate solutions based on the following Ans¨atze for Nonadiabatic Trajectory Surface Hopping (TSH) dynamics the total wavefucntion [ROKS: N. L. Doltsinis, D. Marx, PRL, 88,166402(2002)] Born- 1 C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95,163001(2005) (r, R, t) j (r; R)⌦j (R, t) E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98,023001(2007) Huang ! Xj t Ehrenfest i (r, R, t) (r, t)⌦(R, t)exp E (t0)dt0 Time dependent potential energy surface approach ! el  ~ Zt0 based on the exact decomposition: (r, R, t)=⌦(R, t)(r, t). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105,123002(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation

t Ehrenfest i (r, R, t) (r, t)⌦(R, t)exp Eel (t0)dt0 ! ~ t0  Z We start from the polar representation of the nuclear wavefunction

Inserting this representation of the total wavefunction into the molecular td Schr¨odinger equation and i multiplying from the left-hand side by ⌦⇤(R, t)andintegratingoverR we get ⌦(R, t)=A(R, t)exp S(R, t)  ~ 2 @(r, t) ~ 2 ˆ i~ = i (r, t)+ dR ⌦⇤(R, t)V (r, R)⌦(R, t) (r, t) where the amplitude A(R, t)andthephaseS(R, t)/~ are real functions. @t 2me r Xi Z Inserting this representation for ⌦(R, t)andseparatingtherealandtheimaginarypartsonegets

2 e2Z for the phase S in the classical limit ~ 0 ˆ e ! where V (r, R)= iConservation of energy has also to be imposed through the condition that d Hˆ /dt 0. h i ⌘ and the trajectories of the corresponding (quantum) mechanical systems. Note that both the electronic and nuclear parts evolve according to an average potential generated by the other component (in square brakets). These average potentials are time-dependent and are responsible for the feedback interaction between the electronic and nuclear components.

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Ehrenfest dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics - the nuclear equation Ehrenfest dynamics - the nuclear equation

The identiﬁcation of S(R, t)withthe”classical”action,deﬁnesapoint-particledynamicswith @S 1 1 2 ˆ = M S dr ⇤(r, t) el (r, R)(r, t) @t 2 r H Hamiltonian, Hcl and momenta Z X P = S(R). rR The solutions of this Hamiltonian system are curves (characteristics)inthe(R, t)-space, which are extrema of the action S(R, t)forgiveninitialconditionsR(t )andP(t )= S(R) . 0 0 rR |R(t0) Newton-like equation for the nuclear trajectories corresponding to the HJ equation

dP = dr ⇤(r, t) ˆel (r, R)(r, t) dt r H Z

Ehrenfest dynamics

@(r; R, t) ˆ i~ = el (r; R)(r; R, t) @t H Instead of solving the ﬁeld equation for S(R, t), ﬁnd the equation of motion for the M R¨ = ˆ (r; R) I I rI hHel i corresponding trajectories (characteristics).

Ehrenfest dynamics @ 1 The identiﬁcation of S( , t)withthe”classical”action,deﬁnesapoint-particledynamicswith 2 R i~ k (r, t)= r k (r, t)+ve↵[⇢, 0](r, t) k (r, t) @t 2me r Hamiltonian, Hcl and momenta ¨ ˆ P = S(R). MI RI = I el (r; R) rR r hH i The solutions of this Hamiltonian system are curves (characteristics)inthe(R, t)-space, which are extrema of the action S(R, t)forgiveninitialconditionsR(t )andP(t )= S(R) . 0 0 rR |R(t0) Newton-like equation for the nuclear trajectories corresponding to the HJ equation

dP = dr ⇤(r, t) ˆel (r, R)(r, t) dt r H Z

Ehrenfest dynamics - Densityfunctionalization (k :KSorbitals)

@ 1 2 i~ k (r, t)= r k (r, t)+ve↵[⇢, 0](r, t) k (r, t) @t 2me r M R¨ = E[⇢(r, t)] I I rI

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Ehrenfest dynamics Mixed quantum-classical dynamics Ehrenfest dynamics Ehrenfest dynamics and mixing of electronic states Ehrenfest dynamics and mixing of electronic states

Ehrenfest dynamics

@(r; R, t) ˆ Ehrenfest dynamics i~ = el (r; R)(r; R, t) @t H el i~c˙k (t)=ck (t)Ek i~ cj (t)Dkj MI R¨I = I ˆel (r; R) r hH i Xj

¨ 1 2 el Consider the following expansion of (r; R, t)inthestatic basis of electronic wavefucntions MI RI = I ck (t) E r | | k k=0 k (r; R) X { } 1 where (r; R, t)= ck (t)k (r; R) @ @R @ ˙ ˙ k=0 Dkj = k j = k j = R k j = R dkj X h | @t | i h | @t @R | i h |r| i · 2 The time-dependency is now on the set of coecients ck (t) ( ck (t) is the population of state Thus we incorporate directly nonadiabatic e↵ects. { } | | k). Inserting in the Ehrenfest’s equations...

We can “derive” the following trajectory-based solutions:

Nonadiabatic Ehrenfest dynamics dynamics I. Tavernelli et al., Mol. Phys., 103,963981(2005).

Adiabatic Born-Oppenheimer MD equations

Nonadiabatic Bohmian Dynamics (NABDY) B. Curchod, IT, U. Rothlisberger, PCCP, 13,32313236(2011) Nonadiabatic Trajectory Surface Hopping (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88,166402(2002)] C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95,163001(2005) el E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98,023001(2007) i~c˙k (t)=ck (t)E i~ cj (t)Dkj k Xj

1 Time dependent potential energy surface approach M R¨ = c (t) 2E el I I rI | k | k based on the exact decomposition: (r, R, t)=⌦(R, t)(r, t). k=0 X A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105,123002(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Born-Oppenheimer approximation Born-Oppenheimer approximation

Born- 1 (r, R, t) j (r; R)⌦j (R, t) Huang ! Xj Born- 1 (r, R, t) j (r; R)⌦j (R, t) Huang ! Xj Electrons are static.Useyourfavoriteel.str.method.

In this equation, j (r; R) describes a complete basis of electronic states solution of the For the nuclei,insertthisAnsatz into the molecular time-dependent Schrödingerequation time-independent Schrödingerequation: @ Hˆ (r, R, t)=i~ (r, R, t) @t ˆ (r; R) (r; R)=E (R) (r; R) Hel j el,j j After left multiplication by k⇤(r; R)andintegrationoverr,weobtainthefollowingequation(we R is taken as a parameter. used = ): h j | i i ij Eigenfunctions of êl (r; R)areconsideredtobeorthonormal,i.e. j i = ij . H h | i 2 ~ 2 1 @ I + Eel,k (R) ⌦k (R, t)+ Dkj ⌦j (R, t)=i~ ⌦k (R, t) " 2MI r # @t XI Xj

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Born-Oppenheimer approximation Born-Oppenheimer approximation

2 ~ 2 @ I + Eel,k (R) ⌦k (R, t)+ Dkj ⌦j (R, t)=i~ ⌦k (R, t) " 2MI r # @t 2 I j ~ 2 X X Dkj = k⇤(r; R) I j (r; R)dr " 2MI r # Z XI 1 Equation for the nuclear “wavepacket”, ⌦(R, t), dynamics. + k⇤(r; R)[ i~ I ] j (r; R)dr [ i~ I ] MI r r XI ⇢Z Eel,k (R)representsapotentialenergysurfaceforthenuclei.

If we neglect all the Dkj terms (diagonal and o↵-diagonal), we have the Born-Oppenheimer Important additional term : Dkj ! NONADIABATIC COUPLING TERMS approximation.

2 ~ 2 2 @ Dkj = ⇤(r; R) j (r; R)dr ~ 2 k 2M rI I + Eel,k (R) ⌦k (R, t)=i~ ⌦k (R, t) Z " I I # " 2MI r # @t X XI 1 + k⇤(r; R)[ i~ I ] j (r; R)dr [ i~ I ] Mainly for ground state dynamics or for dynamics on states that do not couple with others. MI r r XI ⇢Z (Back to nonadiabatic dynamics later).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Born-Oppenheimer approximation: the nuclear trajectories Born-Oppenheimer approximation: the nuclear trajectories

2 2 @Sk ~ 1 I Ak 1 1 2 2 = MI r MI I Sk Ek ~ 2 @ @t 2 Ak 2 r + Eel,k (R) ⌦k (R, t)=i~ ⌦k (R, t) I I 2M rI @t X X " I I # X @Ak 1 1 1 2 = M I Ak I Sk M Ak Sk @t I r r 2 I rI XI XI Using a polar expansion for ⌦ (R, t), we may ﬁnd a way to obtain classical equation of motions k Dependences of the functions S and A are omitted for clarity (k is a index for the electronic for the nuclei. i state; in principle there is only one state in the adiabatic case). ⌦k (R, t)=Ak (R, t)exp Sk (R, t) .  ~ We have now a time-dependent equation for both the amplitude and the phase. Ak (R, t)representsanamplitudeandSk (R, t)/~ aphase. Since we are in the adiabatic case there is only one PES and the second equation becomes Further: insert the polar representation into the equation above, do some algebra, and separate trivially a di↵usion continuity equation. real and imaginary part, we obtain an interesting set of equations: @Sk The nuclear dynamics is derived from the real part ( @t ). This equation has again the form of a classical Hamilton-Jacobi equation.

2 2 2 2A @Sk ~ 1 Ak 1 1 2 @Sk ~ 1 I k 1 1 2 I = M r M S E = MI r MI I Sk Ek I I I k k @t 2 A 2 r @t 2 Ak 2 r I k I I I X X X X @Ak 1 1 1 2 = MI I Ak I Sk MI Ak I Sk @t r r 2 r The classical limit is obtained by taking1: ~ 0 XI XI !

@Sk 1 1 2 = M I Sk Ek @t 2 I r I X These are the classical Hamilton-Jacobi equation and S is the classical action related to a particle.

t t S(t)= L(t0)dt0 = E (t0) Epot(t0) dt0 kin Zt0 Zt0 ⇥ ⇤ The momentum of a particle I is related to

v I I S = pI = r MI Instead of solving the ﬁeld equation for S(R, t), ﬁnd the equation of motion for the corresponding trajectories (characteristics). 1Caution! This classical limit is subject to controversy... Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Therefore, taking the gradient, Ehrenfest dynamics @S 1 k 1 2 @(r; R, t) ˆ J = J M I Sk + J Ek i~ = el (r; R)(r; R, t) r @t 2 r I r r H I @t X MI R¨I = I ˆel (r; R) k r hH i and rearranging this equation using J Sk /MJ = v J , we obtain the (familiar) Newton equation: r Explicit time dependence of the electronic wavefunction. d k MJ v = J Ek dt J r

Born-Oppenheimer dynamics In Summary: ˆ (r; R) (r; R)=E el (R) (r; R) Adiabatic BO MD Hel k k k ¨ el ˆ ˆ el MI RI = I Ek (R)= I k el k el (r; R)k (r; R)=E (R)k (r; R) r r h |H | i H k mink ¨ el ˆ MI RI = I Ek (R)= I k el k The electronic wavefunction are static (only implicit time-dependence). r rmink h |H | i

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Mixed quantum-classical dynamics Adiabatic Born-Oppenheimer dynamics Mean-ﬁeld vs. BO MD (adiabatic case) Tarjectory-based quantum and mixed QM-CL solutions

We can “derive” the following trajectory-based solutions:

Method Born-Oppenheimer MD Ehrenfest MD Nonadiabatic Ehrenfest dynamics dynamics adiabatic MD (one PES) nonadiabatic MD (mean-ﬁeld) I. Tavernelli et al., Mol. Phys., 103,963981(2005). Ecient propagation of the nuclei Get the “real” dynamics of the electrons Adiabatic nuclear propagation Propagation of nuclei & electrons t 10-20 a.u. (0.25-0.5 fs) t 0.01 a.u. (0.25 as) Adiabatic Born-Oppenheimer MD equations ⇠ ⇠ Simple algorithm Common propagation of the nuclei and the electrons implies Nonadiabatic Bohmian Dynamics (NABDY) more sophisticated algorithms B. Curchod, IT*, U. Rothlisberger, PCCP, 13,32313236(2011) Trajectory Surface Hopping (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88,166402(2002)] Exact quantum dynamics? C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95,163001(2005) Can we derive “exact” quantum equations of motion for the nuclei? E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98,023001(2007) (without taking the classical limit ~ 0?) ! Time dependent potential energy surface approach based on the exact decomposition: (r, R, t)=⌦(R, t)(r, t). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105,123002(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Nonadiabatic Bohmian dynamics Mixed quantum-classical dynamics Nonadiabatic Bohmian dynamics NABDY: “exact” trajectory-based nonadiabatic dynamics NABDY: “exact” trajectory-based nonadiabatic dynamics

Using From the NABDY equations we can obtain a Newton-like equation of motion (using the HJ (r, R, t)= j1 j (r; R)⌦j (R, t) j i ⌦j (R, t)=Aj (R, t)exp Sj (R, t) definition of the momenta Sj (R, t)=P ) P ~ r ⇥ ⇤ in the exact time-dependent Schrödingerequation for the nuclear wavefucntion we get 2 d R j M = E (R)+ j (R, t)+ Dij (R, t) j 2 r el Q i 2 2 A ( , t) (dt ) @Sj (R, t) 1 2 el ~ j R = S (R, t) + E (R) r h X i r j j @t 2M 2M Aj (R, t) X X 2 2 where j (R, t)isthequantumpotentialresponsibleforallcoherence/decoherence ~ Ai (R, t) i ~ Ai (R, t) i Q + D (R) e d (R) r e ji < ji < “intrasurface” QM e↵ects, and ( , t)isthenonadiabatic potential responsible for the 2M A (R, t) M A (R, t) j R Xi j h i X,i=j j h i D 6 amlpitude transfer among the di↵erent PESs. ~ Ai (R, t) i + d (R) Si (R, t) e M ji A ( , t) r = ,i=j j R h i X6 For more informations see: and B. Curchod, IT, U. Rothlisberger, PCCP, 13,3231–3236(2011) @Aj (R, t) 1 1 2 = A (R, t) S (R, t) A (R, t) S (R, t) r j r j j r j @t M 2M X X NABDY limitations ~ i ~ i + D (R)Ai (R, t) e d (R) Ai (R, t) e 2M ji = M ji r = i h i ,i=j h i Many numerical challenges X X6 1 i d (R)Ai (R, t) Si (R, t) e , Instabilities induced by the quantum potential M ji r < ,i=j h i X6 Compute derivatives in the 3N dimensional(R3N )configurationspace 1 where both Sj (R, t)andAj (R, t)arerealfieldsand = (Si (R, t) Sj (R, t)). ~ Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Nonadiabatic Bohmian dynamics Mixed quantum-classical dynamics Trajectory Surface Hopping Tarjectory-based quantum and mixed QM-CL solutions TSH nonadiabatic MD

We can “derive” the following trajectory-based solutions: There is no derivation of TSH dynamics. The fundamental hypothesis beyond TSH is that it is Nonadiabatic Ehrenfest dynamics dynamics possible to design a dynamics that consists of: I. Tavernelli et al., Mol. Phys., 103,963981(2005). ↵ propagation of a “quantum” amplitude, Ck (t), associated to each PES, k Adiabatic Born-Oppenheimer MD equations ↵ 1 ↵ (r, R, t)= Ck (t)k (r; R) Xk Nonadiabatic Bohmian Dynamics (NABDY) (the label ↵ is to recall that we have a di↵erent contribution from each di↵erent trajectory). B. Curchod, IT, U. Rothlisberger, PCCP, 13,32313236(2011) classical (adiabatic)timeevolutionofthenucleartrajectoriesonadiabaticstatessolution Trajectory Surface Hopping (TSH) dynamics [ROKS: N. L. Doltsinis, D. Marx, PRL, 88,166402(2002)] of the Schr¨odingerequation for the electronic sub-system. Each trajectory ↵ carries a C. F. Craig, W. R. Duncan, and O. V. Prezhdo, PRL, 95,163001(2005) phase, C ↵(t), associated to each surface k. E. Tapavicza, I. Tavernelli, U. Rothlisberger, PRL, 98,023001(2007) k transitions (hops) of the trajectories between electronic states according to a stochastic algorithm, which depends on the nonadiabatic couplings and the amplitudes C ↵(t). Time dependent potential energy surface approach k based on the exact decomposition: (r, R, t)=⌦(R, t)(r, t). See also: J. Tully, Faraday discussion, 110,407(1998). A. Abedi, N. T. Maitra, E. K. U. Gross, PRL, 105,123002(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Tully’s surface hopping - How does it work? Tully’s surface hopping - How does it work?

Equation of motion for the amplitudes: The main claim of TSH is that, the collection of a large enough set of independent trajectories ˙ ↵ ↵ ˙ ↵ ↵ gives an accurate representation of the nuclear wave packet i~C (t)= C (t)(Hkj i~R d ) k j · kj j ↵ ↵ ↵ X CL ↵ ↵ Nk (R , dV , t ) 1 ↵ ↵ 2 ↵ 2 ⇢k (R , t )= ⌦k (R , t ) Ck,R↵,t↵ Ntot dV ⇠ | | ⇠ | | Switching algorithm:

Inserting In the fewest switches algorithm, the transition probability from state j to state k in the time ↵ 1 ↵ (r, R, t)= Ck (t)k (r; R) interval [t, t + dt] is given by: Xk t+dt ↵ ↵ ↵ ↵ ↵ into the molecular time-dependent Schr¨odingerequation and after some rearrangement, we [Ck (⌧)Cj ⇤Hkj (⌧)] [Ck (⌧)Cj ⇤(⌧)⌅kj (⌧)] g ↵(t, t + dt) 2 d⌧ = < jk ⇡ C ↵(⌧)C ↵ (⌧) obtain: Zt j j ⇤ ↵ ˙ ↵ ↵ ˙ ↵ ↵ ↵ ↵ i~Ck (t)= Cj (t)(Hkj i~R d kj ) where ⌅ (⌧)=R˙ d (⌧). A hop occurs between j and k if · kj · kj Xj ↵ ↵ with H = (r; R) ˆ (r; R) . gjl < ⇣ < gjl , kj h k i|Hel | j i el l k 1 l k In the adiabatic representation, we have Hkk = Ek and Hkj =0. X X where ⇣ is generated randomly in the interval [0, 1].

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Tully’s surface hopping - Summary Tully’s surface hopping - Examples

1D systems Tully’s surface hopping

˙ ↵ ↵ ˙ ↵ ↵ i~C (t)= C (t)(Hkj i~R d ) k j · kj Xj M R¨ = E el (R) I I rI k ↵ ↵ gjl < ⇣ < gjl , l k 1 l k X X Some warnings:

1 Evolution of classical trajectories (no QM e↵ects – such as tunneling – are possible). J.C. Tully, J. Chem. Phys. (1990), 93, 1061 2 Rescaling of the nuclei velocities after a surface hop (to ensure energy conservation) is still amatterofdebate.

3 Depending on the system studied, many trajectories could be needed to obtain a complete statistical description of the non-radiative channels. For more details (and warnings) about Tully’s surface hopping, see G. Granucci and M. Persico, JChemPhys126,134114(2007).

Trajectory-based nonadiabatic molecular dynamics Trajectory-basedVery good nonadiabatic agreement molecular with dynamics exact nuclear wavepacket propagation.

Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Tully’s surface hopping - Examples Tully’s surface hopping - Examples

1D systems 1D systems

J.C. Tully, J. Chem. Phys. (1990), 93, 1061

On the right: population of the upper state (k=mom) exact TSH • – Landau-Zener

Trajectory-basedVery good nonadiabatic agreement molecular with dynamics exact nuclear wavepacket propagation. Trajectory-basedVery good nonadiabatic agreement molecular with dynamics exact nuclear wavepacket propagation. Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Tully’s surface hopping - Examples Comparison with wavepacket dynamics

1D systems Butatriene molecule: dynamics of the radical cation in the ﬁrst excited state.

J.C. Tully, J. Chem. Phys. (1990), 93, 1061

On the right: population of the upper state exact TSH • JPCA,107,621 (2003)

Trajectory-basedVery good nonadiabatic agreement molecular with dynamics exact nuclear wavepacket propagation. Trajectory-based nonadiabatic molecular dynamics

Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Comparison with wavepacket dynamics Comparison with wavepacket dynamics

Butatriene molecule: dynamics of the radical cation in the ﬁrst excited state.

Nuclear wavepacket dynamics on ﬁtted potential energy surfaces (using MCTDH with 5 modes) . Reappearing of the wavepacket in S after 40fs. 1 ⇠

JPCA,107,621 (2003)

CASSCF PESs for the radical cation (Q14:symmetricstretch,✓:torsionalangle).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics Mixed quantum-classical dynamics Trajectory Surface Hopping Mixed quantum-classical dynamics Trajectory Surface Hopping Comparison with wavepacket dynamics Comparison with wavepacket dynamics

On-the-ﬂy dynamics with 80 trajectories On-the-ﬂy dynamics with 80 (crosses). trajectories.

Trajectories are not coming back close Trajectories are not coming back close to the conical intersection. to the conical intersection.

What is the reason for this discrepancy? What is the reason for this discrepancy? The independent trajectory The independent trajectory approximation?, i.e. the fact that approximation?, i.e. the fact that trajectories are not correlated? trajectories are not correlated? (Or it has to do with di↵erences in the (Or it has to do with di↵erences in the PESs?) PESs?)

JPCA,107,621 (2003) JPCA,107,621 (2003)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TDDFT-based trajectory surface hopping TDDFT-based trajectory surface hopping

1 Ab initio molecular dynamics Tully’s surface hopping - On-the-ﬂy dynamics Why Quantum Dynamics? Nuclear quantum e↵ects Tully’s surface hopping 2 Mixed quantum-classical dynamics ˙ ↵ ↵ ˙ ↵ ↵ Ehrenfest dynamics i~C (t)= C (t)(Hkj i~R d ) k j · kj Adiabatic Born-Oppenheimer dynamics Xj Nonadiabatic Bohmian dynamics M R¨ = E el (R) Trajectory Surface Hopping I I rI k

3 TDDFT-based trajectory surface hopping g ↵ < ⇣ < g ↵ , Nonadiabatic couplings in TDDFT jl jl l k 1 l k X X 4 TDDFT-TSH: Applications Photodissociation of Oxirane What about the electronic structure method for on-the-ﬂy dynamics? We need:

Oxirane - Crossing between S1 and S0 Potential energy surfaces MR-CISD, LR-TDDFT, semiempirical, ... ! Forces on the nuclei MR-CISD, LR-TDDFT, semiempirical methods, ... . 5 TSH with external time-dependent ﬁelds ! Local control (LC) theory Nonadiabatic coupling terms MR-CISD, LR-TDDFT (?), semiempirical methods, ... . ! Application of LCT

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TDDFT-based trajectory surface hopping TDDFT-based trajectory surface hopping Tully’s surface hopping - On-the-ﬂy dynamics Tully’s surface hopping - On-the-ﬂy dynamics

Tully’s surface hopping Tully’s surface hopping

˙ ↵ ↵ ˙ ↵ ↵ ˙ ↵ ↵ ˙ ↵ ↵ i~C (t)= C (t)(Hkj i~R d ) i~C (t)= C (t)(Hkj i~R d ) k j · kj k j · kj Xj Xj

M R¨ = E el (R) M R¨ = E el (R) I I rI k I I rI k

↵ ↵ ↵ ↵ gjl < ⇣ < gjl , gjl < ⇣ < gjl , l k 1 l k l k 1 l k X X X X

What about the electronic structure method for on-the-ﬂy dynamics? We need: What about the electronic structure method for on-the-ﬂy dynamics? We need: Potential energy surfaces MR-CISD, LR-TDDFT, semiempirical, ... Potential energy surfaces MR-CISD, LR-TDDFT, semiempirical, ... ! ! Forces on the nuclei MR-CISD, LR-TDDFT, semiempirical methods, ... . Forces on the nuclei MR-CISD, LR-TDDFT, semiempirical methods, ... . ! ! Nonadiabatic coupling terms MR-CISD, LR-TDDFT (?), semiempirical methods, ... . Nonadiabatic coupling terms MR-CISD, LR-TDDFT (?), semiempirical methods, ... . ! !

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT TDDFT-TSH: Applications

Nonadiabatic couplings with LR-TDDFT? 1 Ab initio molecular dynamics Why Quantum Dynamics? Nuclear quantum e↵ects Nonadiabatic coupling vectors are deﬁned in terms of electronic wavefunctions: 2 Mixed quantum-classical dynamics Ehrenfest dynamics ˆ k (R) R el j (R) Adiabatic Born-Oppenheimer dynamics dkj = k (R) R j (R) = h |r H | i h |r | i E (R) E (R) j k Nonadiabatic Bohmian dynamics Trajectory Surface Hopping The main challenge is to compute all these quantities as a functional of the ground state 3 TDDFT-based trajectory surface hopping electronic density (or equivalently, of the occupied Kohn-Sham orbitals). Nonadiabatic couplings in TDDFT d d [⇢] kj ! kj 4 TDDFT-TSH: Applications Photodissociation of Oxirane 2 Di↵erent approaches for the calculation of d0j [⇢] are available . Oxirane - Crossing between S1 and S0 Here we will use the method based on the auxiliary many-electron wavefunctions. 5 TSH with external time-dependent ﬁelds Local control (LC) theory 2V. Chernyak and S. Mukamel, J. Chem. Phys. 112, 3572 (2000); R. Baer, Chem. Phys. Application of LCT Lett. 364, 75 (2002); E. Tapavicza, I. Tavernelli, and U. Rothlisberger, Phys. Rev. Lett. 98, 023001 (2007); C. P. Hu, H. Hirai, and O. Sugino, J. Chem. Phys. 127, 064103 (2007). Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TDDFT-TSH: Applications TDDFT-TSH: Applications Protonated formaldimine Protonated formaldimine The protonated formaldimine is a model compound for the study of isomerization in rhodopsin Computational details chromophore retinal. Isolated system In addition to the ground state (GS), two excited electronic states are of interest: LR-TDDFT/PBE/TDA

1 S : ⇡ (low oscillator strength) SH-AIMD 1 ! ⇤ 2 S : ⇡ ⇡ (high oscillator strength) 50 trajectories (NVT) each of 100 fs. 2 ! ⇤ ⇠

PRL, 98, 023001 (2007); THEOCHEM, 914, 22 (2009).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TDDFT-TSH: Applications TDDFT-TSH: Applications Protonated formaldimine Protonated formaldimine

Typical trajectory Protonated formaldimine as a model compound for the study of the isomerization of retinal.

Photo-excitation promotes the

system mainly into S2.

Relaxation involves at least 3 states:

S0 (GS), S1 and S2.

[E. Tapavicza, I. T., U. Rothlisberger, PRL, 98,

023001 (2007); THEOCHEM, 914, 22 (2009)]

PRL, 98, 023001 (2007); THEOCHEM, 914, 22 (2009).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TDDFT-TSH: Applications TDDFT-TSH: Applications Protonated formaldimine Protonated formaldimine

↵ Nonadiabatic couplings = R˙ d ↵ States population kj · kj

PRL, 98, 023001 (2007); THEOCHEM, 914, 22 (2009). PRL, 98, 023001 (2007); THEOCHEM, 914, 22 (2009).

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TDDFT-TSH: Applications TDDFT-TSH: Applications Protonated formaldimine Protonated formaldimine

States population - Average over many trajectories. Geometrical modiﬁcations Dashed line = CASSCF result.

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TDDFT-TSH: Applications TDDFT-TSH: Applications Oxirane - Crossing between S1 and S0 Protonated formaldimine Oxirane

Comparison with experiment and model calculations Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation In addition to the isomerization channel, intra-molecular proton transfer reactions was can occur. + observed (formation of CH3NH ).

H2 abstraction is also observed in some cases. Structures and life times are in good agreement with reference calculations performed using high level wavefunction based methods.

Figure: Mechanism proposed by Gomer and Noyes

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TDDFT-TSH: Applications Oxirane - Crossing between S1 and S0 TDDFT-TSH: Applications Oxirane - Crossing between S1 and S0 Oxirane Oxirane

Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation Oxirane has interesting non-radiative decay channels, during which ring opening and dissociation can occur. can occur. Computational details Isolated system LR-TDDFT/PBE/TDA SH-AIMD 30 trajectories (NVT) each of 100 fs. ⇠

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TDDFT-TSH: Applications Oxirane - Crossing between S1 and S0 TSH with external time-dependent ﬁelds

The photophysics of solvated Ruthenium(II) tris-bipyridine 1 Ab initio molecular dynamics

2+ Why Quantum Dynamics? [Ru(bpy)3)] dye: photophysics 2+ [Ru(bpy)3)] dye: Singlet state dynamics Nuclear quantum e↵ects

N 2 Mixed quantum-classical dynamics

N y N Ehrenfest dynamics z x N Adiabatic Born-Oppenheimer dynamics N N Nonadiabatic Bohmian dynamics Trajectory Surface Hopping 2+ [Ru(bpy)3)] dye: Solvent structure 3 TDDFT-based trajectory surface hopping Nonadiabatic couplings in TDDFT

2+ [Ru(bpy)3)] dye: triplet state dynamics 4 TDDFT-TSH: Applications Photodissociation of Oxirane

Oxirane - Crossing between S1 and S0

5 TSH with external time-dependent ﬁelds Local control (LC) theory [M.E. Moret, I.T., U. Rothlisberger, JPC B, 113,7737(2009);IT,B. Application of LCT Curchod, U. Rothlisberger, Chem.Phys., 391,101(2011)]

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TSH with external time-dependent fields TSH with external time-dependent fields TSH with external time-dependent fields TSH with external time-dependent fields

Addition of an external ﬁeld within the equations of motion of TSH: Short summary of the theory

The interaction Hamiltonian between the electrons and the td electric ﬁeld is

ˆ e Hint = A(r i , t) pˆi 2me c · Xi

where A(r, t)isthe(classical)vectorpotentialoftheelectromagneticﬁeld,pˆi is the momentum operator of electron i, e is the electron charge, me is the electron mass, and c is the speed of light.

Startegy Remark The idea is to induce electronic excitations through the direct interaction with the time-dependent We are in the dipole approximation and therefore we do not need TDCDFT. (td) electric ﬁeld instead of “artiﬁcially” promote the system into one of its excited states. IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81,052508(2010) Method:extendedTSHnonadiabaticdynamics.

IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81,052508(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TSH with external time-dependent fields TSH with external time-dependent fields External field within TSH E↵ect of an electromagnetic field - Lithium fluoride

Di↵erent excitations can be obtained, depending on the polarization vector of the laser pulse. It can be shown (Phys. Rev. A 81 052508 (2010)) that through the coupling with the td electric ﬁeld, Tully’s propagation equations acquire an additional term Electronic structure of LiF

˙ ↵ ↵ ˙ ↵ ↵ A0 ↵ i!t Ground state - ⌃ i~CJ (t)= CI (t)(HJI i~R d JI + i!JI ✏ µJI e ) · c · symmetry (GS) . XI with First excited state (doubly A0(t) i!JI µ = J Hˆint I degenerate) - ⇧ symmetry c · JI h | | i (S1). and where A (t)=A ✏e i!t is the vector potential of the external td electric ﬁeld, 0 0 Second excited state - ⌃ symmetry (S ). µ = e ˆr 2 JI h J | i | I i Xi Avoided crossing between GS and S2 is the the transition dipole vector, and !JI =(EJ EI )/~.

Note that Tully’s hops probability should be modiﬁed accordingly. IT, B. Curchod, U. Rothlisberger, Phys. Rec. A 81,052508(2010)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TSH with external time-dependent ﬁelds Local control (LC) theory TSH with external time-dependent ﬁelds Application of LCT

Local control theory Application: Photoexcitation of LiF in the bound state S2

Control is achieved by tuning the temporal evolution of E(t)inawaytomaximizethe population of a target state.

Using the TSH for the total molecular wavefunction

↵ 1 ↵ (r, R, t)= CJ (t)J (r; R) XJ for a given trajectory ↵,thepopulationtimeevolutionsimpliﬁesto

↵ ↵ ↵ ˙ (t)= 2E (t) [C ⇤µ C (t)] PI = J JI I XJ It is now evident that choosing a ﬁeld of the form Test system: LiF.

↵ ↵ The aim is to excite the molecule selectively into the S2 bound-excited-state starting from the E(t)= [CI (t)CJ ⇤µIJ )] = ground state. XJ will ensure that (t)alwaysincreasesintime. The dynamics is initiated in the ground state and the population is transferred from the GS into PI S2 through direct coupling with an electron magnetic pulse (pi-piulse or local controlled pulse).

T. J. Penfold, G. A. Worth, C. Meier, Phys. Chem. Chem. Phys. 12,15616(2010). B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84,042507(2011) B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84,042507(2011)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT E↵ect of a generic polarized pulse LC pulse: ecient population transfer and stable excitation

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT Comparison with wavepacket propagation (MCTDH) Local control on proton transfer (in gas phase)

B.F.E Curchod, T. Penfold, U. Rothlisberger, IT, PRA, 84,042507(2011)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT Local control on proton transfer Local control on proton transfer

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT Local control on proton transfer (with one water molecule) Local control on proton transfer (with one water molecule)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT Local control on proton transfer (comparison) Local control on proton transfer (freq. vs. time)

gas phase with one water molecule

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics

TSH with external time-dependent ﬁelds Application of LCT TSH with external time-dependent ﬁelds Application of LCT Local control on proton transfer (mode analysis) Recent review on TDDFT-based nonadiabatic dynamics

ChemPhysChem,14,1314(2013)

Trajectory-based nonadiabatic molecular dynamics Trajectory-based nonadiabatic molecular dynamics